Hurwitz Number Fields David P. Roberts, U. of Minnesota, Morris Akshay Venkatesh, Stanford University

Transcription

1 Hurwitz Number Fields David P. Roberts, U. of innesota, orris Akshay Venkatesh, Stanford University Notation: NF m (P) is the set of isomorphism classes of degree m number fields ramified only within P and with associated Galois group all of A m or S m.. Sets NF m (P) and mass heuristics. First example 3. fields in NF 5 ({, 3, 5}) 4. Generalities I: Families 5. Generalities II: Specialization 6. fields in NF ({, 3, 5}) 7. Concluding remarks

2 . Sets NF m (P) and mass heuristics. Let F P (m) = NF m (P). Some known cases: P {} {} 3 {, 3} {, 3, 5} {, 3, 5, 7} {, 3, 5, 7, } {, 3, 5, 7,, 3} E.g., NF 5 ({, 3}) = {Q[x]/f i (x)} i=,...,5 with f (x) = x 5 x 4 + 4x 3 6x + f (x) = x 5 x 4 + x 3 + 4x 5x + f 3 (x) = x 5 x 4 x 3 + 6x 6x + 6 f 4 (x) = x 5 x 3 4x 9x 4 f 5 (x) = x 5 x 4 + 4x 3 x + x A local-global mass heuristic lets one predict F P (m). For example, it predicts F {,3,5,7} (5) 556 while in fact F {,3,5,7} (5) = 465. It seems reasonable to expect that the prediction is asymptotically correct as one goes down a column.

3 But what about going to the right on a row, i.e. the behavior of F P (m) for fixed P and increasing m? The literal predictions of the mass heuristic are as follows in some examples: 3 log F P (m) 5 5 5,3,3,5,3,5,7 This might lead one to expect that, for any fixed P, no matter how large, F P (m) is eventually zero. This may be indeed be correct for small P. For example, the largest m for { F{} (m) which F {,3} (m) { m = m = 64. is known to be nonzero is

4 However... For Γ a non-abelian finite simple group, let P Γ be the set of primes dividing Γ. Note that the only such P Γ with P Γ 3 are {, 3, p} with p {5, 7, 3, 7}. Define P to be large if P contains some P Γ and small otherwise. So if P or P then P is small. This talk is about a systematic (and rather classical!) construction of what we call Hurwitz number algebras. Their ramifying primes are very well controlled and evidence points strongly to Galois groups being generically the full alternating or symmetric groups on the degree. Accordingly we now think, Conjecture. For any fixed large P, the number F P (m) can be arbitrarily large.

5 . First example. Consider polynomials in C[y] of the form g(y) = y 5 + by 3 + cy + dy + e. The four critical values are given by the roots of the resultant r(t) = Res y (g(y) t, g (y)). Explicitly, this resultant works out to r(t) = 35t 4 +5(3bc e)t 3 + ( 8b 5 9b 3 d + 85b c 5bce + bd +5c d + 875e ) t ( 8b 5 e 36b 4 cd + 8b 3 c 3 9b 3 de + 85b c e + 8b cd 35bc 3 d 565bce + bd e + 54c 5 + 5c de 8cd e 3) t + ( 8b 5 e 7b 4 cde + 6b 4 d 3 + 6b 3 c 3 e 4b 3 c d 9b 3 de +85b c e + 56b cd e 8b d 4 63bc 3 de + 44bc d 3 375bce 3 + bd e + 8c 5 e 7c 4 d + 5c de 6cd 3 e + 56d e 4).

6 To find all polynomials g(y) whose critical values are, say,,,,, we need to solve r(t) = 35(t + )t(t )(t ) for (b, c, d, e). The twenty-five e that arise are the roots of F (e) = e , which is irreducible over Z. A better defining polynomial for Q[e]/F (e) is f(x) = x 5 5x 4 + 5x 3 5x 38x + 9x 45x 9 88x x x x x x x 4734x x x x x x x x x x 5897x Its field discriminant is known a priori to be of the form ± 3 5 and is in fact D = = Not much is known a priori about the Galois group of f(x). It works out to A 5.

7 Some pictures illustrating the situation: Five real polynomials with critical values,,, The preimage of [, ] under all twenty-five polynomials

8 3. fields in NF 5 ({, 3, 5}) Our specialization polynomial (t + )t(t )(t ) can be replaced by any quartic polynomial with leading coefficient and discriminant divisible only by, 3, and 5. Via changes of coordinates, most cases are covered by the family s(u, v; t) = t 4 6ut 8ut 3uv. The corresponding moduli polynomial F (u, v; e) has 45 terms with coefficients averaging 37 digits. A small search gets 3 pairs (u, v) which keep ramification in {, 3, 5}:

9 A D B - - C E Top: Specialization points (u, v) for F (u, v; e). Bottom: the discriminant locus (thick) and special lines (thin).

10 Over each of the special lines, the defining equation can be much simplified. E.g. over u = v it becomes f AB (u, x) = 4( u)(x + ) ( 79x 8 486x 7 7x 6 8x 5 + 5x 4 + 8x 3 557x + 96x 576 ) u(x ) 4 x 9. Computation shows that all 3 specialization points give A 5 or S 5 fields. The behavior of the exponents a, b, c in D = ± a 3 b 5 c is very constrained: p p p 5

11 4. Generalities I: Families. In general, a Hurwitz number algebra is indexed by its parameter, H = (λ,..., λ l ; Z,..., Z l ; ). The parameter for our first example was H = (, 5; {,,, }, { }; S 5 ). In general, the λ i are partitions of a given positive integer n, the Z i are disjoint Gal(Q/Q)- stable subsets of P = C { }, and is a transitive degree n permutation group of the form Γ or Γ. with Γ non-abelian simple. Let X H be the set of degree n covers of P, ramified only over Z i, with local ramification partition λ i for all t Z i, and global monodromy group. Then Gal(Q/Q) acts naturally on X H and the corresponding Hurwitz number algebra is K H. Its primes of bad reduction are within the primes of bad reduction for Z i and the primes dividing.

12 Replacing each Z i by its size z i gives the corresponding familial parameter: h = (λ,..., λ l ; z,..., z l ; ). Each h gives a chain of varieties Y h n Xh P X h m Uz,...,z l. One starts with a focus on degree n covers of P and ends with a complicated degree m cover X h of the very simple variety U z,...,z l. One can cut down dimensions by three via the natural P GL (C) action. In our degree twentyfive case h = (, 5; 4, ; S 5 ), the u-v plane is an essential slice of the five-dimensional variety U 4,. Over this slice, the cover X h is given by the equation F (u, v; e) =.

13 Important invariants of families are n, the degree of the original cover z, the number of ramification points g, the genus of the original cover m, the degree of the moduli cover Log N 7 n 5 n g Dots correspond to families

14 z = n g λ λ λ 3 λ 4 λ 5 m µ z = 3 n g λ λ λ λ λ 3 m µ a b 6 6. z= 4 n g λ λ λ λ λ m µ z = 3 n g λ λ λ λ λ m µ z = 5 n g λ λ λ λ λ m µ z = n g λ λ λ λ 3 λ 4 m µ a b a b z= n g λ λ λ λ λ 3 m µ a b a b ab 6 4 6ab Five-point families with genus and low degree

15 5. Generalities II: Specialization. For each partition z, we have a base-stack U z over Z which contains our specialization points (after quotient by P GL ). We are interested in their points U z (Z P ). These are very explicit sets. Examples: U,,, (Z P ) = {t Z P : t Z P } U,,, (Z {,3} ) = S 3 {, 3, 4, 9} ( elements) U 3, (Z P ) = {j-invs for ECs over Z P } U 3, (Z {,3,5} ) = 44 It s easy to produce elements of these sets. In favorable cases, one can prove that there are no more elements, e.g. for the cases U 3 c b a (Z {,3,5} ). For example, U 5, 4 (Z {,3,5} ) = 3, 93, 3, 4,.

16 For the conjecture, it is important to prove that for all large P, the sets Uz (ZP ) can be arbitrarily large. In fact this is true for all nonempty P, via cyclotomic polynomials and their near-relatives. Example: the roots of an irreducible near-relative of a cyclotomic polynomial of degree 5 = and discriminant of the form ±. This polynomial gives one of many systematically constructed points in U43768,,,(Z{})

17 6. fields in NF ({, 3, 5}). To construct these fields: I. Construct family belonging to h = (3 b c, 3, 4, ;,,, ; S 6 ). This procedure starts with consideration of g(y) = ay3 (y b) (y c) y + dy + e with (a, b, c, d, e) chosen so that the critical values are,, and the two roots of (t + ut + v). The result is a polynomial F (u, v; b) with 96 terms. II. Plug in the 947 elements (u, v) in the specialization set U,,, (Z {,3,5} ). Each gives a degree polynomial of field discriminant of the form ± a 3 b 5 c. To support the conjecture, we would like many of these polynomials to be irreducible with Galois group all of A or S.

18 The specialization set U,,, (Z {,3,5} ) is For all 947 specialization points, the Galois group of F (u, v; b) is all of A or S. Even degenerations of our polynomial F (u, v; b) have degrees which are large enough to pose computational challenges. For example F ( t, t ; x) = (3x x+) 5 f 3 (t, x) 3 f 48 (t, x) These degenerations give 4-point NFs.

19 7. Concluding Remarks. A. Starting with degree n families of alle and others, involving the simple groups Γ = P SL (7), P SL (8), P SL (),, we get degree m families, still with A m and S m as desired, but now with other bad primes, e.g. {, 3, 7}. B. There seems hope for predicting the ramification of Hurwitz number algebras in terms of the placement of the specialization point in the relevant specialization set U z (Z P ). C. So what does the sequence of F P (m) = NF m (P) look like for P large? We don t know, but perhaps something in the spirit of...,,,,...,,,,,,...,,,,,,...